Pergamon
Chemical En~lineerin9 Science, Vol. 52, No. 20, pp. 3527-3542, 1997
PIh S0009-2509(96)00157-7
i' 1997 Elsevier Science Lld. All rights reserved
Printed in Great Britain
0oo9-2509/97 $17.00 + 0.00
Internal mass transfer in sintered metallic
pellets filled with supercritical fluid
Frank Stiiber, St6phan Julien, F. Recasens*
Department of Chemical Engineering, ETS Enginyers Industrials de Barcelona,
Universitat Polit6cnica de Catalunya, Diagonal 647 08028 Barcelona, Spain
(Received 18 November 1996; accepted in revised form 18 April 1997)
Abstract--The shrinking core model has been used in some cases to describe kinetics of
extraction of seeds with a supercritical fluid as solvent, despite the fact that seed geometry may
be quite irregular and internal walls may strongly affect diffusion and the liquid-fluid-solid
equilibria. Consequently, the values of model parameters (effective diffusivity, particle-fluid
mass-transfer coefficient) hardly, if ever, prove meaningful or have a limited value. In the
present work, the extraction of benzene, toluene, ethylbenzene, and 1, 2-dichlorobenzene from
shallow packed beds of macroporous pellets of well-defined geometries, are studied both
experimentally and theoretically using the shrinking core model. Sintered metallic pellets,
formed by powder metallurgy in two sizes, were used. These were bronze cylinders with either
sealed ends or open ends. For a bed a few particles thick, the differential reactor approximation
is acceptable. With this assumption, the extraction models contain only three parameters--a
liquid-gas partition coefficient, an external mass-transfer coefficient, and an intraparticle or
effective diffusivity. The former two were determined in separate experiments (solubility) or
taken from previous studies (mass transfer coefficient), so that the effective diffusion coefficient
is the only model parameter to be fitted. The procedure adopted eliminates the usual uncertainties regarding geometry, conditions at the boundaries, and axial mass dispersion in the bed.
A careful look at experimental results allows to distinguish between equilibrium-limited and
rate-limited extraction process. By fitting extraction data for the rate-limited runs to model
solutions, values of the effective diffusivity were calculated. Abnormally low (less than unity)
tortuosity factors which decrease with temperature were obtained. ~'~ 1997 Elsevier Science Ltd
Keywords: Supercritical carbon dioxide; mass-transfer; diffusion; macropores; sintered metals.
INTRODUCTION
Supercritical fluid extraction (SCFE), particularly
when carbon dioxide is the fluid, has proved attractive
as compared with other separation processes. A field
where it has gained attention is the extraction of
porous matrices. Large mass transfer rates and the
capacity for intrusion into granular structures, has
turned the SCF into a powerful solvent in applications such as soil cleanup (Brady et al., 1987; Laitinen
et al., 1994; Montero et al., 1996), activated carbon
regeneration (Tan and Liou, 1988, 1989; Recasens
et al., 1989; Srinivasan et al., 1990; Madras et al.,
1994), extraction of seeds and roots (Jones, 1991;
Sovovfi et al., 1994; Roy et al., 1996), and impurity
desorption from granular materials (Chouchi et al.,
1995).
The desorption of solute from porous materials and
its evaporation into the SCF is rather complex and
involves various steps. The solute trapped in the por-
* Corresponding author.
ous matrix by physical or chemical forces must first be
dissolved in the fluid at an interior site. Once dissolved, the solute must diffuse through the pores to
the outer particle surface. When adsorption phenomena are present along the pore, diffusion may also
proceed in the adsorbed state within the pore walls
resulting in the so-called surface migration, by a type
of on-and-off adsorption-desorption (Smith, 1981).
Surface diffusion has been invoked by Knaff and
Schliinder (1987) and Lai and Tan (1993) for interpreting extraction data in particles with SCF-filled macro- and micropores, respectively. Once the dissolved
solute molecules have reached the solid-fluid interface
they move across the stagnant boundary layer to the
bulk fluid and are transported by convection out of
the extractor.
For the case of liquid solutes, the initial steps of the
extraction mechanism are even more complex due to
the relatively large solubility of the compressed gas in
the liquid, which causes a significant reduction in the
density and viscosity of the liquid solution. This situation leads to what has been described by Barton
(1992) as capillary drying. When SC carbon dioxide is
3527
3528
F. Stiiber et al.
absorbed by the organic solute trapped in the pores,
the solution swells and becomes less viscous due to
the presence of dissolved carbon dioxide. The wicked,
organic-rich solution may spread out over the external surface of the particle, where it is subject to
evaporation into the bulk fluid. After a certain fraction of the solution has evaporated the external surface is no longer completely wetted and intraparticle
diffusion may become the main transport limiting
step.
Thus, mass transport properties undergo significant
changes during the course of the extraction process.
Initially, mass transfer control is likely to reside in the
liquid organic solution-SCfluid film, with an effective
interfacial area approaching the total external surface
area of the porous particle (Barton et al., 1992; Stiiber
et al., 1996b). When the external surface is no longer
fully wetted, the stagnant liquid-fluid film turns into
a dense gas solution-solid one and the effective mass
transport area reduces to that of pore mouths, Recently, Stiiber et al., (1996b) have measured and correlated the particle-to-fluid mass transfer coefficients
applicable in the presence of natural convection mass
transfer. Also, Puiggen6 et al. (1996) published correlations for characterization of the free liquid-to-fluid
mass transfer.
Another feature is that a rapid evaporation rate of
the solute film during the initial period of extraction
may lead to a liquid-fluid interface temperature below
that of the bulk gas, for cases where evaporation
is endothermic (above the crossover pressure). The
interface temperature would tend to reach the
adiabatic-saturation temperature for the system.
A few degrees may affect liquid-vapour equilibrium
widely, particularly near the critical pressure of the
mixture.
In general, modelling the extraction process of solid
matrices shows increasing interest for scaleup calculations and for process optimization. Brunner (1984),
Roy et al. (1996) and Goto et al. (1996) have intensively modelled the extraction of seeds. A short review is
given by Goto et al. These authors assimilate seed
grains to spherical shrinking-core particles, in order
to simulate the kinetics of evaporation of bound
moisture, irrespective of actual particle geometry. In
this regard the shrinking core concept is useful in the
prediction of extraction time (Goto et al., 1996), and
the dissolution of solid naphthalene from sintered
metal rods, hence with a well-defined geometry (Knaff
and Schliinder, 1987). The shrinking-core model
(SCM) has been used also for the removal of binder
from ceramic green bodies with SC carbon dioxide,
and for the extraction of toluene and dichlorobenzene
from sintered metallic cylinders (Stiiber et al., 1996a).
The SCM has been presented by King and Catchpole
(1993) in connection with physico-chemical design
data together with typical values of the parameters.
Most of the extraction data corresponded to integral fixed-bed reactor operation, with explicit allowance for non-ideal residence time distribution, either
in terms of axial dispersion (Madras et al., 1994; Goto
et al., 1996; Roy et al., 1996) or by use of a radial
distribution of residence times based on non-uniform
flow conditions (Sovovh et al., 1994).
Nishikawa et al. (1991) compared the SCM with the
bulk solid diffusion model and concluded that both
could account (within 6% difference) for the extracted
fractions vs time for binder removal from ceramics.
On the other hand, Barton et al. (1992) mentioned
that the fitted effective diffusivities for the SCM for the
extraction of binder from ceramic change as much as
4 orders of magnitude. Instead, they successfully fit
their data by a model in which the binder is brought
to the surface by capillarity. However, it should be
mentioned that the aim of their experiments was to
extract just a portion of the binder (usually below
45%). Furthermore, it may be questionable to apply
the SCM for extraction where internal diffusion is not
likely to be the rate-limiting step.
Our objective in this paper was to develop and
apply the shrinking-core concept for particles with
well defined external geometries and boundary conditions. The pellets used were bronze cylinders prepared
by powder metallurgy, having 1 and 2 cm, respectively, equivalent diameters. The impregnating solutes
were benzene, toluene, ethylbenzene and 1,2-dichlorobenzene. The loaded pellets were packed into shallow
beds (2-3 particles thick) for extraction. Extraction
data allowed the measurement of solute diffusivity
into the SCF-filled porous space.
EXPERIMENTAL METHODS
A detailed description of experimental procedures
has been given elsewhere (Stiiber et al., 1996b).
A sketch of the extraction setup and a scope of the
experimental conditions are shown in Fig. 1 and
Table 1, respectively. During a standard extraction
run, subcooled liquid CO2 was pumped to the operating pressure through the extractor cylinder (3 cm ID)
via a heater to control operating temperature. Extraction pressure was kept around + 1% of the desired
value. Temperature fluctuations were less than
+ 0.5 K. Actual CO2 flowrate and the total amount
passed could be measured by use of a rotameter and
a wet gas meter with temperature reading. As required
%
~ 1
c
'iii!i
,)
Fig. 1. Experimental apparatus.
Supercritical fluid extraction
Table 1. Experimental conditions
1
F
"
0,9
Extraction
Carbon dioxide flowing
mode
Temperature, K
Pressure, MPa
Reynolds number
Schmidt number
Solutes extracted
3529
Upflow
310-360
8-20
20-85
1.5 10
Benzene,toluene, ethylbenzene,
1,2 Dichlorobenzene
Pellets*
Small cylinders (bronze, open
ends) large cylinders (bronze,
sealed ends)
small pellets
large pellets
Packing properties
Layers of pellets in bed
2
3
Height, cm
2
5
Bed cross section, cm 2
7
7
Pellet dp x Lp, mm
8 x 8.2
20 × 11.6
Equivalent diameter, cm
1
2
Specific area of packing,
m2/m 3
450
80
Pellet porosity, %
20.5
25
Average pore size, m × 10 6
18-20
14 20
0,8
/
0,7
*
u 0,6
~.~
t~ 0,5
;~
0,4
0,3
/
/
if
0,2
0,1
DCB - C02 L_
P = 12 MPa
T = 52°(2
Re=14
"-k
0
0
1000
2000
3000
4000
t(s)
la}
1,6
*Prepared by powder metallurgy at AMES (Sant Vicen~
dels Horts, Barcelona).
1,2
0,8
by the dynamic nature of the experiments, the tubing
between extractor exit and detection was kept minimal. Also, the initial setting of pressure to set point was
made rapidly to avoid undue extraction at startup.
The runs were performed with benzene (Bz), toluene
(Tol), ethylbenzene (EB), and 1,2-dichlorobenzene
(DCB) as impregnating solutes for the macroporous
metal cylinders (see Table 1). Before an experiment,
the extractor was preheated without pellets to shorten
the time required by the system to reach stable equilibration (1-3 min). Two types of experiments were
done to measure either extraction kinetics or to
measure equilibrium solubility by operating at a very
small gas flowrate. These latter quasi-equilibrium runs
were based on that described by Sovovfi et al. (1994).
In extraction kinetics runs, the extractor cell was
packed with a shallow bed of pellets placed between
sections of inert packing (1 mm diameter glass beads
and glass wool). Depending on conditions and progression of extraction about 10-15 samples were
taken at 0.5-20 rain time intervals. Solute recovery
was usually better than 93-95%. Replicate runs were
done to verify reproducibility. This was found to be
excellent.
Figure 2(a) shows a typical extraction profile, in
terms of the experimental data points and the model
prediction (continuous line). As can be seen, mass
fraction vs time curves are generally composed of
a nearly linear initial portion (fast extraction period)
and a flat end tail (slow extraction period) connected
by a transition section zone. The figure shows also the
mean concentration profile in the bulk fluid. The
gas-phase concentration is useful for checking model
0,4
0
(h}
0,01
0,02
0,03
kg C02
0,04
0,05
Fig 2. (a) Experimental (points) and calculated (solid line)
extraction and relative gas-phase concentration profiles for
a typical run (DCB extraction from small pellets. (b) Extracted mass of toluene vs mass of COz passed at very small
flowrate as in quasi-equilibrium runs, P = 12MPa,
T = 52°C.
predictions of the complete extraction profiles. Mean
concentrations, Co, can be established from solute
weight recovered, time elapsed, volume of gas passed
during the same time interval, and an estimate of the
fluid density at extractor conditions.
To determine the solubilities of the solutes in SC
CO2, quasi-equilibrium runs were carried out at some
selected conditions. In these, the extraction cell was
filled with a long section of 3 mm glass beads packed
between sections of cotton wool and dry glass beads
(1 mm diameter). The first glass beads and the cotton
were wetted with liquid solute and gently squeezed.
The CO2 flowrate was kept very low 0.1-0.3 1/min
(1 bar and 20°C) to insure that the fluid leaving the
cell is always saturated. A typical quasi-equilibrium
run is represented in Fig 2(b) for the case of toluene.
The extracted mass fraction as a function of the
3530
F. Stfiber et al.
amount of CO2 passed is given by a straight line, thus
confirming that the fluid leaves the cell always
saturated. Saturation was further verified by changing
the CO2 flow rate three times (0.1 to 0.3 1/min). The
measured extracted fractions are still on the same
straight line in Fig. 2(b), indicating also saturation at
a higher flowrate.
Equations (3)-(5) indicate that in the present model
extraction takes place by vaporization at the liquidSCF interphase within the pores followed by diffusion
of solute to the outer surface of the particle.
For a pellet whose core radius is re, the extracted
mass fraction is
X(t, z) =
M O D E L L I N G THE EXTRACTION PROCESS
The following models describe the extraction of
porous particles initially wetted with liquid solute.
After a certain fraction of the external moisture has
been carried away by the flowing fluid, drying rate
becomes affected by diffusion from an internal intact
liquid core through the pores and to external particle
surface. The rate depends also on particle shape.
Three particle geometries are considered: cylinders
with sealed ends, spheres, and cylinders with porous
ends. In the shrinking-core model, when the solute
concentration in the liquid core is much higher than
in the pore fluid solution as in our case, a sharp
liquid-fluid boundary develops and the liquid core
recedes regularly as extraction progresses. In the
model the following assumptions are made: (1) P, T, u,
and ~B are uniform and constant throughout the bed.
Hence, solubility is constant; (2) the bed is isotropic,
so diffusional coefficients (Kg, De) are independent of
direction; (3) the time change of solute concentration
in the pores is small compared to its change with the
radial position (pseudo-steady-state approximation)
(Est6vez and Smith, 1985); (4) at the interface of the
liquid core and the fluid, extraction proceeds at equilibrium, so the fluid is saturated with solute. Since the
fluid phase is a dilute solution, equilibrium between
liquid and fluid is linear, hence it can be characterized
by a partition coefficient, K.
The following fundamental expressions are derived
for a packed bed of cylinders with sealed ends. The
solute mass balance for fluid assumed in plug flow, is
(see Notation)
OC° u OC° 2~v(1
+
_
- en) Ko[(C,)R _ Col.
Or
8z
R~B
_
_
(6)
Boundary and initial conditions are
Ci(r = R) = (Ci)R,
Ci(r = rc) = C*
CJt, z = 0) = 0,
Co(t = 0, z) = 0
r<(t = 0, z) = R.
(ci)R =
(9)
(10)
De
ln('r</R)
evRK°
Finally, substituting this expression into eqs (1) and
(2), as well as letting the rate given by eq. (2) equal to
that given by eq. (4), provides two equations for the
change of fluid-phase concentration and the change of
core radius. These are
ocj + uOC
__ o _
0t
2
0z
[" 1 - e.B'~
Re't~)evln
KoD e
0c)
R
RK o_De
x ( C * - Co)
(1)
Or¢
K
(11)
RKoD e
Ot
(2)
x (c~' - co).
0Ci
r 8r t r ~ - r ) = e P & -
(3)
with the pseudo-steady-state approximation, the rhs
of eq. (3) is zero. Diffusional flux at core surface:
( OCi /~ .... .
- ONo__t= 2xr~LD~\--~r
77
f Ore\
-- n p L L r c ~ v l - - - - I
\
Equations (11) and (12) are the governing equations
for the extraction of a bed of cylinders with sealed
ends with fluid in plug flow. Using a similar procedure, the governing equations are found for other types
of particle geometry. Thus, for spheres of radius R,
they are
OCg
&
2
ot ) . . . . '
(12)
(4)
Rate of dissolution of liquid core:
ON
(8)
DeC*
In(re~R)
epRK°C°
Pore diffusion:
De 0 { 0C~'~
(7)
Using the pseudo-state approximation in eq. (3) together with the boundary conditions, eq. (7), it is possible to obtain the intraparticle concentration profile
Ci(r). Differentiation of the latter with respect to r, and
evaluation of the derivative at rc provides the gradient
required in eq. (4). Combining that derivative with
eqs(2) and (4), the unknown concentration at the
external surface is obtained as
External mass transfer to fluid:
ON
& = 2x%RLKoE(CDR - Co].
-
1
(5)
uOCo
8z
3c. ( l - c . ~
= -R " \ - T Z J
× (Cg* - Cg)
K,D e
~ - i-- ) + Do
~,RKo(7
(13)
Supercritical fluid extraction
K
C* frc'2V
~r c
f
~t
KoD ~
[ R _ 1 ) + D~]
x (C* - Co).
(14)
For cylinders with open ends, the equations are derived applying the superposition principle, based on
the fact that the ends behave like a flat plate. These are
combined with the lateral diffusion for a cylinder with
sealed ends. The total flux to the fluid phase is the sum
of 3 fluxes, 2 from the flat ends, and another from the
lateral surface of the cylinder. The governing equations for cylinders of length L, are found to be
OCg
8t
uOCo
8z
2
I
x
/1 - e~\
1
L
ee ~ RK o ln(rffR) - D~
+~(21~_ 1
2\ L
RK o
~
--R
differential conditions, as there is just one parameter
to fit (De), provided that solubility and external mass
transfer data are available. We show later that even
for differential conditions, operation near the critical
point is to be avoided because saturation occurs in
a bed length of a few particles.
For very shallow beds, a few particles thick, the
differential reactor approximation is valid much like
in drying rate studies. For such beds, the equations for
Co, rc and l< become ordinary differential equations
with time as the variable. The space derivatives are
substituted by the corresponding finite difference in
terms of the residence time of fluid in the bed. For the
differential problem the initial conditions are kept
the same. Furthermore, all pellets in the bed are in the
same state of extraction, so eq. (6) will provide the
total fraction of bed extracted and X becomes independent of z. Then, the rates of extraction in terms of
the core dimensions are
for cylinders with sealed ends:
dX
dt -
D~
c
~t
~
K
C*rc
m
~1___5= K
c?t C*
RKaD~
C* Co)
f r<\
21¢( o ~eln t ~ ) R K u - De
RKgD~
L ["21~ ~
(22)
and for cylinders with open ends:
(17)
where r~ is the radius of the remaining cylindrical core,
and 21c is the distance between the two flat bed fronts
progressing from the ends to the centre of the pellet
(2lc = length of the cylindrical core). Additional
boundary and initial conditions are required for
I¢ (t, z) as well as for Ci at the new boundaries. These
are
(18)
Ci(r, 1 = + L/2) -= (Ci)R
l<(t = o, z) = +_
R 3 \ dt//
dX
dt -
(C* - Co)
1 = + 1D = C*o
[21)
(16)
r~
ev~ t--~ - 1/ R K o - D~ --~
Ci(r,
2rcCdr<~
R 2 \ dt j
for spheres:
(15)
~r
m
3531
(19)
L
(20)
Solution of models and limiting cases
A new assumption is necessary regarding the length
of the bed. The above equations are for an integral
packed bed where two kind of pitfalls may occur.
First, after travelling 10 or 12 particles diameters
along the bed, the gas may become saturated with
solute, hence making rate data impossible to measure.
The second problem refers to axial dispersion. Using
an intergral bed, allowance for mass dispersion is
necessary for the length of our extraction cell. Then,
a new parameter is needed for the model. On the other
hand, it seems attractive to operate the bed under
dX
at -
2rfl,.(drc~_ rZ~ (die "]
R2L \ dt J RZL \ dt J
(23)
These equations can be integrated together with the
corresponding conservations equations for each geometry. Numerical solution was performed using the
Gear algorithm (DIVPAG, IMSL, 1987) which is a
robust method for stiffly coupled equations.
The denominators ofeqs (11)-(17) clearly show two
resistances in series together with a driving force factor in the numerator. One resistance is diffusion in the
pores, which is variable with time, and the particlefluid mass transfer resistance which is constant. The
internal diffusion resistance is negligible at the beginning of extraction and maximum at longer times. We
can use the Biot number for characterizing the ratio of
internal to external mass transfer gradients, although
it may not be strictly so because pore diffusion resistance is variable. For limiting values of Bi, the model
equations reduce to simplified expressions. Two interesting cases arise.
The first case is for Bi ~ O, i.e., external mass-transfer resistance is limiting. In this case the extraction
rates are given by a similar equation for the 3 geometries studied,
dX_bKKo(l_
dt
Co)
~*o*
(24)
where factor b depends on the geometry, b = 3/R for
a sphere; b = 2/R for cylinder with sealed ends, and
b = (1/R + l/L), for a cylinder with open ends. As
3532
F. Sttiber et al.
expected, the extraction rate depends only on the
mass transfer coefficient, and is independent of diffusivity. See eq. (24). The second case is for limiting
internal diffusion, or B i - o oo. The rates for spheres
and cylinders with sealed ends, respectively, are given
by
dt - R 2 e p ( R / r c - 1) 1 -
dt -
R e epln(rc/R)
(26)
1-
.
(27)
For cylinders with open ends there is a contribution
from the lateral cylinder surface and another from the
ends. These are
c t / l ....... t -
RE
~
evln(rc/R~) 1 -- ~-~
(~X~
~t / ........ t
2Kr2
R 2L2 [_
F
De
(1
(28a)
Ca
(28b)
which depend only on intraparticle diffusivity, as expected.
In practise, Biot numbers will be intermediate between Bi = 0 and Bi = oo. The sensitivity of model
parameters was studied as a function of the Blot
number. Table 2 shows the effects of De, Kg and K (at
different Biot numbers) on the time required for 50%
extraction of small pellets impregnated with DCB. It
is observed clearly that for intermediate and large
Biot numbers (Bi > 250) the sensitivity of the extraction time to De is large relative to the sensitivity to K a.
This is the expected featured from the meaning of
large Biot numbers, that is when internal resistance is
controlling. However, for small values of Bi ( ~ 5) the
sensitivities to K o and De are about the same. As it will
be presented later, Biot numbers calculated with the
fitted parameters are in the range of Bi = 14-45,
Table 2. Predicted times, tso, for 50% recovery of DCB from
small pellets (T = 313 K, P = 20 MPa)
Bi = 5
Kg, m/s
1×10 -6
t5 o.s
3000
K = 0.3, De =
De, m2/s
tso. s
10 -9
Bi = 5
--
Effect of Kg
Bi = 25
Bi = 250
5×10 -6
5×10 -5
1300
670
Bi = 2500
5x10 -4
600
m2/s
Effect of De
Bi = 25
Bi = 250
10 -s
10 -9
160
670
Bi = 2500
10-to
3 600
K =0.3, Kg = 5 × 10 -s m/s
K
t 5o.s
Effect of K
Bi = 250
1
180
De = 10 -9 m2/s, K a = 5 x 10 -5 m/s.
Bi = 250
0.3
670
Bi = 250
0.1
1600
making the accuracy of De dependent on the accuracy
in the prediction of Kg.
Elucidating the controlling step
At low Biot number and far from equilibrium
(C* <<Cg), eq. (24) suggests that the rate of extraction
will be approximately constant, limited only by the
external mass transfer coefficient. Therefore, the
amount extracted will change linearly with time. On
the other hand, this will never be observed if internal
diffusion is the dominant resistance. Limitation in the
pores would lead to a non-linear variation of X with t,
because the diffusion path, R - re, is not constant
during the process, see eqs (26)-(28).
A constant extraction rate may also occur in a situation where the two mass transfer steps are very rapid,
so that extraction takes place at equilibrium, the bulk
fluid being saturated with solute. Then the extraction
rate is constant, equal to the product of solubility
times the fluid flowrate. In this case, the rate of extraction is independent of geometry and depends only on
flowrate. The time, t*, required for extracting a bed of
length, B, is obtained as
t* - (1 - eB)Bqo
(29)
u~BC*
where qo is the initial concentration of solute on the
solid, and u is the interstitial velocity. Equation (29)
ignores the unsteady state periods, and it provides the
minimum extraction time.
Equation (29) provides a check for saturation. It
allows discrimination of the two cases where a constant extraction rate is observed (external mass transfer limitation vs equilibrium limitation). It is required
only to compare the observed time with that calculated with eq. (29). If both times are close to each
other, it is likely that the overall process is determined
by the solvent capacity of the fluid. In the case of
external mass transfer control, the actual extraction
time is likely to be much larger than the minimum, t*.
EVALUATION OF SYSTEM PROPERTIES
The governing equations describing the extraction
process contain three parameters: the external mass
transfer coefficient (Kg), the solute solubility or partition coefficient (K), and the effective diffusivity in the
pellet, which is related to molecular diffusivity
through a suitable tortuosity factor, i.e., D e = De,p/z.
In order to get more physical insight, a number of
adjustable parameters, approximate solubilities and
mass transfer coefficients were taken from independent measurements. Molecular diffusivities were
taken from contrasted experimental data as well as
reliable correlations.
Solubilities
The solubilities expressed in mole fraction of benzene (Bz), toluene (Tol) and ethylbenzene (EB) in SC
carbon dioxide were experimentally determined in the
pressure and temperature ranges of 8-11.5 M P a and
3533
Supercritical fluid extraction
45-6Y~C. For dichlorobenzene (DCB) VLE had been
characterized in a previous work (Stiiber et al., 1996a).
The method used here was a quasi-equilibrium extraction as described earlier (Experimental Methods). Results are reported in Tables 3 and 4 for all conditions.
The first step was to check the precision of our
method with toluene in SC CO2 at selected conditions
of pressure and temperature. The values for Tol were
compared with the rigorous equilibrium data (Fink
and Hershey, 1990) and those from VLE calculations
using the Peng-Robinson equation-of-state (PREOS, Sandier, 1989), see Table 3. As shown in the
table, our results agree reasonably well with the equilibrium data of Fink and Hershey. At lower temperature (51-52°C) there are deviations of about 15-20%,
but these vanish at 80-81°C. The same trend is observed for the calculated mole fraction from the PREOS using an interaction parameter k12 = 0.077.
Table 4 summarizes both experimental and calculated
solubilities for Bz, Tol, and EB in SC CO2 for ten
operating conditions used in the extraction runs.
Thus, for the system Bz/CO2, k12 = 0.0770, the same
as that reported by Sandler (1989). For the other
systems, Tol/CO2 and for EB/CO2, the same value
k~2 = 0.0770 was find appropriate. For the case of
D C B / C O z , Stiiber et al. (1996a) reported a value of
k~2 = 0.1175 based on a previous work from this
department (Puiggen6 et al., 1996).
Measured and calculated values of mole fractions
agree very well for the case of Bz/CO2, with devi-
Table 3. Solubilities of toluene in C O 2 a s measured by Fink
and Hershey (Fink and Hershey, 1990), our experimental
data and values calculated with the Peng-Robinson equation of state (PR-EOS)
T ('C)
51
65
80
P, MPa
8.2
9.0
9.0
10.0
10.0
11.6
Exptl
y* (toluene)
F&H
PR-EOS
0.0156
0.0185
0.0202
0.0280
0.0263
0.045
0.123
0.016
--0.0252
0.0454
0.0115
0.0150
0.0173
0.023
0.0256
0.0378
For PR-EOS estimate, k~2=0.077 (Sandier, 1989) was
used.
ations less than 5%. For the systems Tol/CO2 and
EB/CO2, the deviations are somewhat larger but still
acceptable (10% and about 20%, respectively). As
expected, solubilities decrease significantly with molecular weight for a series of homologous compounds.
Thus, Bz > Tol > EB. However, at constant pressure
(9 MPa), an increase in temperature of 15°C does not
affect the solubility very much (Table 4). It is also
interesting to note that the EB/CO 2 system clearly
exhibits retrograde solubility. This was also confirmed later in the extraction experiments, where extraction rate is seen to decrease with temperature.
Molecular diffusivities
Molecular diffusivities are required to evaluate the
tortuosity factors. Experimental values of diffusivities
of Bz, Tol, and EB in SC CO2 are reported in the
literature (Suhrez et al., 1993; Lee and Holder, 1994;
Swaid and Schneider, 1979). These data were compared to the values predicted with the correlation of
Catchpole and King (1994) specifically developed for
various solvents and solutes at near-critical conditions.
A typical plot of experimental and calculated diffusivities vs pressure, at constant T = 40°C, for the
Bz-CO2 system is given in Fig. 3. The values of
Suhrez et al. (1993) were extrapolated to pressures
lower than 15 MPa. The correlation of Catchpole and
King predicts very well the experimental values
(Suhrez et al., and Swaid and Schneider). At pressures
above 10 MPa, deviations are negligible. In the nearcritical region the correlations give values that may be
up to 20% larger. But given the steep increase of
molecular diffusivity in the critical region, the fitting is
acceptable. The same trend is observed for the other
systems (Tol, EB) and for different temperatures. We
can conclude that in the absence of experimental data
at SC conditions, the correlation of Catchpole and
King is sufficiently accurate for estimating molecular
diffusivity.
Diffusivity decreases sharply with pressure in the
vicinity of the critical pressure of CO2, up to 10 MPa.
Then it is nearly constant with P. This would mean
that during extraction of a porous material, diffusion
in the pores would be enhanced at pressures near the
critical pressure of the solvent. The same is true also
for the mass transfer coefficient. On the other hand,
Table 4. Measured and predicted solubilities of benzene and ethylbenzene in carbon
dioxide
P (MPa)
8
9
9
9
T (°C)
44
51
58
65
Exp
Calc
y (Toluene)
Exp
Calc
0.018
0.029
0.027
0.030
0.017
0.020
0.029
0.032
-0.018
-0.020
y (Benzene)
For all binary systems, k~ = 0.0770.
0.010
0.015
0.016
0.017
y (Ethyl benzene)
Exp
Calc
0.008
0.015
0.012
0.012
0.006
0.012
0.009
0.002
3534
F. Stiiber et al.
I
•
--
i
J
Swaid & Schneider (1979)
Catchpole & King (1994)
_
0,8
× Su~ez etal.(1993)
P = 8MPa
/
Te432"O5°C _ Bz /
"~
!
"
X
0,6
<
t_
0,4
0,2
i
0
10
20
30
40
300
600
900
1200
1500
t (s)
P Oll~)
Fig. 3. Comparison of measured diffusivitiesfor benzene in
SC CO2 (Sufirezet al., 1993;Swaid and Schneider, 1979)with
the prediction given by the Catchpole-King (1994) correlation along the 40°C isotherm.
Fig. 4. Extraction profiles of different SC systems studied
near the critical pressure P = 8 MPa (Reynolds number,
Re < 25 and T = 42°C.
solubility is still very small at relatively lower pressures. Therefore, the overall extraction is expected to
be limited by the saturation concentration of solute in
the fluid and not by transport properties.
RESULTSAND DISCUSSION
M a s s transfer coefficients
Solution of the shrinking-core model at zero time
depends only on two parameters, the partition coefficient, K, and mass transfer coefficient K0, and is
independent of De. Knowledge of the solubility (see
above), and measuring the initial extraction rate made
it possible to measure KÜ for the DCB-COz and
ToI-CO2 systems. We found (Stiiber et al., 1996b)
that external mass transfer in packed beds occurred
by a combination of natural and forced mass transfer
convection flows. The effect of natural convection
depended on whether the natural convective flow was
in the same or opposed direction to main flow. For
upflow operation, the correlations are
ShT
-
Sho = [ShF - (ShN - Sh0)[
(30a)
ShN -- Sho + O.O01(ScGr)°33Sc °'244
(30b)
She = 0.269Re°'S8Sc °'3
(30c)
Sho = 0.3
(30d)
applicable for the following ranges: 8 < R e < 90,
3.5 x 107 < Gr Sc < 10 9, and 1.5 < Sc < 10 (StiJber
et al., 1996b).
The values of Kg are based on an effective area that
corresponds to the external superficial area of particles. As pointed out earlier, both Kg and the effective
area may vary during the course of extraction because
of the effects of capillary drying, specially during the
inital period. However, the changes in Kg and aeff are
in opposite directions (Kg tends to increase, while
aefr tends to decrease), so that the product is about
constant.
The experimental extraction profiles (extracted
mass fraction vs time and mean fluid-phase concentration vs time) for the various binary systems were
fitted with the solution of the shrinking-core model,
using the differential-reactor approximation. The only
adjusted parameter was the intraparticle diffusivity.
Values for the latter were evaluated as a function of
temperature and pressure, for two different pellet
sizes. Extraction runs at pressures slightly above the
critical pressure of CO2 (hereinafter called low-pressure results) exhibited a particular behaviour and are
discussed separately.
Extraction results at low pressure
Figure 4 shows the fraction extracted vs time for the
four systems studied at 8 MPa and temperatures between 43M5-C, Reynolds number R e = 20-25 and
upflow operation. At these conditions all four profiles
are composed by two linear portions connected by a
short discontinuity where slope changes. Only after
80% of the trapped liquid has been extracted, the
profile becomes flat, suggesting that pore diffusion
retards the overall process. It is interesting to note
that this particular behaviour (the longer middle
straight part) tends to disappear when fluid velocity is
increased, thus shortening the time the fluid spends in
the cell. Figure 5 illustrates extraction runs for about
the same conditions (P = 8 MPa, 40°C) but various
Reynolds numbers, which are R e = 25, 50, 84. At
R e = 50, the break point becomes difficult to see, and
at R e ~ 84 it has completely disappeared. In order to
explain these observations, we tested two cases which
may show a constant extraction rate. These are: (a) the
case where external mass transfer is limiting, or, (b)
extraction proceeds at equilibrium (fluid exits the bed
saturated with solute).
3535
Supercritical fluid extraction
3
1
0,9
/ l/_t.=~
0,8
i
0,7
f"
0,6
O
0,5
/
0,4
/
(
/
/
2,5
]p:45oc
\
8 MPa
Re,=25
%
1,5
I;9
0,3
EB
0,2
0,5
I T = 38-40°C]
Bz . , ~ T o l
0,1
0
600
0
3000
6000
9000
12000
(a) E x t e r n a l m a s s transfer limitin9 case. With the
extraction profiles of Fig. 4 (middle lines) we calculated K o using eq. (24) with b for a cylinder with
open ends, which was derived from the model equations for external mass transfer control. The interfacial area is taken to be that of pore mouths and the
fluid-phase concentration is chosen in the range of
C g - 0 to 0.5 C*. The values of K o so obtained,
together with effective diffusivities selected to be 10%
of molecular diffusivities, were used to estimate the
Biot numbers for these experiments. The values of Bi
obtained fall in the range 25-50 (with Cg = 0 or 0.5
C*, respectively), which are inconsistent with limiting
external mass transfer, for which Bi should take up
a value of around unity or less.
(b) Fluid saturation. To check for saturation, we
calculated both the mean fluid concentration, C 9 / C * ,
and the extraction time assuming equilibrium extraction, t*, given by eq. (29). Figure 6 gives the plot of the
relative saturation vs time for EB, Tol, Bz at P = 8
MPa, T---45°C, and R e = 20-25, and Fig. 7 for
D C B - C O 2 at pressures of 8, 12, and 20 MPa. In
Table 5 values of the extraction times, texp, were compared with the minimum ones, t*, for various operating conditions. Figure 6 clearly indicates that at P =
8 M P a the fluid is saturated most of the time. Note
that in the initial region there exists a period of oversaturation which will be addressed later. The saturation period matches well the linear parts of the
corresponding curves of Fig. 4. The same trend is
reflected by the values of t~p and t* at the same
conditions (See Table 5). toxp and t* are close to each
other suggesting fluid saturation. The situation is different for higher pressures (see Fig. 7 and centre of
Table 5) as well as for higher Reynolds number (bottom of Table 5). Increasing pressure results in both
higher solubilities and smaller values of transport
1800
2400
3000
t (s)
t (s)
Fig. 5. Extraction pofiles for the system DCB/CO2/ small
pellets for various Reynolds numbers near the critical pressure P = 8 MPa, T = 42°C.
1200
Fig. 6. Bulk fluid concentrations at bed exit of Bz, Tol, EB at
P = 8 MPa, Re = 20-25 and T = 45°C.
2
t
1,8
DcB-co2
1,6
T = 40 - 50°C
Re = 8 - 20
1,4
-Itu
1,2
!
0,8
"
0,6
0
0
2000
4000
6000
8000
10000
t (s)
Fig. 7. Bulk fluid concentrations of DCB for different pressures, small Reynolds number and T = 40 50°C.
Table 5. Comparison of minimum extraction times, t*, with
observed extraction times, texp, for different operation conditions (small pellets)
Extraction conditions: P = 8.0 MPa, T = 43°C, Re ~ 22
System
DCB EB
Tol
Bz
Observed time, texp, rain
170
50
45
30
Minimum time, t*, min
100
40
30
20
t* calculated with eq. (29)
Extraction of DCB, effect of pressure at T=45"C,
Re ~ 8-20
P, MPa
8.0
12.0
20.0
Observed time, texp, min
175
60
65
Minimum time, t*, min
100
15
5
Extraction of DCB, effect of SCF velocity at T = 4 0 C ,
P = 8 MPa
Reynolds number
24
56
84
Observed time, texp min
175
70
40
Minimum time, t*, min
100
30
10
3536
F. Stiiber et al.
properties. As seen in Fig. 7, the concentrations in the
fluid no longer reach equilibrium and consequently
the value of the extraction time, texp, becomes much
larger than t* (see middle part of Table 5, extraction
of DCB).
In the case of higher Reynolds number (i.e., shorter
residence time) the fluid does not have time to become
saturated. Values of toxp are also larger with Re than
the values of t* (Table 5 bottom, extraction of DCB).
However, the effect of a smaller residence time on
texp/t* is less pronounced than the effect of pressure.
As mentioned earlier, the dynamic profiles of the
mean fluid concentration at low-pressure exhibit an
initial oversaturation peak (see Figs 4 and 7). At t = 0,
due to the effect of capillary drying, pellets are likely
to be covered by a solute-rich wicked liquid layer
overflowing from the pores, thus creating additional
mass transfer area. This liquid film is then subject to
a rapid evaporation, soon after the SC fluid flow is
started. The mass transfer is fast because the evaporation area and gradient are both maximal. Assuming
endothermic evaporation, the temperature of the
interface would be below that of the bulk fluid. On the
other hand, solubility is known to be a strong function
of temperature, particularly near the critical pressure.
Thus, the observed oversaturation as well as the break
point may be a result of too small a solubility when it
is calculated at the bulk fluid temperature. For
example, in the case of DCB at 8 MPa, a liquid
temperature 1.5°C below the bulk gas temperature,
leads to a two-fold increase in solubility, or, C*
(39.5°C) ~ 2C* (41°C), (calculated with PR-EOS, k12
= 0,1175). The other solutes qualitatively follow a
similar trend. In summary, for large mass transfer
rates the interface liquid would reach its adiabaticsaturation temperature. After this rapid evaporation
period, and for longer extraction times, the external
surface is no longer wetted by a layer of liquid and the
heat and mass transfer gradients are reduced, so vaporization effects are significantly reduced and the
liquid temperature approaches that of the bulk fluid.
In conclusion, for the low-pressure runs, at low
Reynolds number, the rate of extraction of Bz, Tol, EB
and DCB are determined by their respective equilibrium solubilities. Since the latter are small, and the
transport coefficients such as K g and D are very high,
the extracting fluid soon reaches the saturation concentration, even in a packed bed only 2-3 pellets
thick. Under these circumstances, the shrinking-core
concept or any other type of kinetic model based on
a finite rate cannot be used to measure transport
parameters.
For higher pressure and larger velocities, extraction
is no longer limited by the capacity of the fluid, and
then the SCM can be applied to measure rate coefficients, effective diffusivity, in particular.
Extraction runs at higher pressures
The data at higher pressures were compared with
model predictions in order to determine values of the
effective diffusivity (De) at various conditions. In
Table 6 the values of De and those of the tortuosity
factor and Biot number are listed together with the
corresponding operating conditions. In particular, the
runs performed with systems of CO2 and either EB,
Tol, or Bz, at constant pressure (9 MPa) and various
Table 6. Effective diffusivities, tortuosity factors and solubilities for use with the shrinking-core
model for the SC CO2 extraction of aromatic hydrocarbons from sintered metal pellets
Solute
Bi
108D*
(mr/s)
108De
(m2/s)
D/De
rt
Ks
52
59
67
17
17
15
4.9
5.7
5.8
3.0
4.5
5.5
1.6
1.2
1.1
0.32
0.25
0.22
0.15
0.05
0.03
9.
9
9
51
57
65
16
13
15
4.6
5.0
5.9
3.1
4.3
5.2
1.5
1.1
1.i
0.31
0.24
0.23
0.04
0.03
0.02
9
9
9
51
56
67
14
16
15
4.2 4.8
5.8
3.2
4.0
5.3
1.3
1.2
1.1
0.27
0.25
0.22
0.03
0.01
0.01
12
12
22
20
20
42
62
53
39
41
45
40
45
25
30
1.2
2.4
1.9
0.9
1.0
0.5
2.0
1.3
0.4
0.4
2.6
1.2
1.5
2.2
2.5
0.48
0.25
0.31
0.45
0.62
0.10
0.03
0.03
0.23
0.25
P
T
MPa
°C
Benzene
9
9
9
Toluene
Ethyl benzene
DCB
Data for Bz, Tol, and EB, are for small pellets (ep = 0.20), except for DCB which is for large pellets
(~p= 0.25)
* Estimated with the Catchpole-King (1994) correlation.
t Tortuosity factor defined in Notation section.
Partition coefficient from quasi-equilibrium runs.
3537
Supercritical fluid extraction
temperatures (51-67°C), were used to study the dependence of D e on T. Experimental extraction profiles
and model predictions are illustrated in Figs 8(a)-(c)
for the 3 solutes and for the highest and lowest ternperatures studied.
Additional extraction data were available for the
D C B - C O 2 system to test the model at different pressures and temperatures, as well as the effect pellet size.
Comparison between the calculated and experimental
extraction profiles are illustrated in Fig. 9a for cylinders of 1 and 2 cm (equivalent diameter), and in
Fig. 9b for pressures of 12 and 20 MPa, at otherwise
the same operating conditions. From Figs 8 and 9 it
can be seen that, in general, model and experiment
agree, after parameter fitting, for the cases tested.
Fitting is good also for the mean fluid-phase concentration profiles which have been skipped in the previous figures for simplicity. The average standard
deviation (average error) is about 10-15% and the
values of the diffusivity are accurate to within + 15%.
For example, the model is able to predict extraction
profiles for small and large pellets at the same operating conditions (P and T) updating and the external
mass transfer coefficient using Stfiber et al.'s (1996b)
correlation using a single value of De = 0.4x
10 -8 m2/s and [see Fig. 9(a) and Table 6].
To further check its validity, the model was initialized with saturation conditions for the fluid phase,
or, Cg (t = 0 ) = C*, thereby simulating the startup
with saturated carbon dioxide. Fluid saturation still
may have some importance in the runs at P = 9 MPa
(Figs 8(a)-(c)). Model values for the extracted fraction
with initial saturation at short extraction times were
found significantly below both the experimental
values and the values calculated with C9(0) = 0. This
indicates that saturation of fluid is no longer important at pressures of 9 MPa or above.
In a second verification, we arbitrarily changed the
values of the partition coefficient and the external
mass transfer coefficient. For example, our De values
were relatively large judged from the point of view of
heterogeneous catalysis and adsorption. Therefore,
we increased K and/or Kg to force D e to take a smaller
value (comparable with those of catalysis) but still
fitting the experimental extraction profiles within
a similar goodness of fit. For the smaller De, the
predicted fluid-phase concentration profiles showed
large deviations, being sometimes 2 times smaller than
the values experimentally observed. Thus, fitting only
the extracted fraction vs time curves may lead to
erroneous values of model parameters. As discussed
above, different sets of parameters may provide the
same goodness-of-fit. In our case, parallel measurement and fitting of fluid phase concentrations was
found useful in selecting meaningful sets of parameters.
The fitted values of De are listed in Table 6 and are
also plotted in Fig. 10, in terms of tortuosity factors as
a function of temperature. For comparison, data
available from SCFE of macroporous matrices are
also shown in Fig. 10. Values for tortuosities in SCFE
1
•
~ 0,6
~ 0,5
.o
0,4
/
x
-"
3<
0,9
0,8
~¢ 0,7 - T ffi6 ~
T ffi52"C
[,1(
o
i/
~ 0,3
,/
0,2
0,1 t
0 ,
0 300 600 900 1200
Bz- C02 ~P = 9 MPa IP.e = 2 0 - 2 5 !
I
I
1500 1800 2100
t (s)
(a)
1
.
.....
0,9
0,8 - - T = 6 6
~¢, 0,7
C ~ . . . ""
/'" T = 50°C
~ 0,6
~O 0,5
©
0,4
~
0,3
0,2
/
Tol - CO2
P = 9 MPa
Re = 20 - 25 -
0,1
0
I
I
I
300 600 900 1200 1500 1800 2100
t (s)
(b)
1
0,9
~¢
~
~
0,8
0,7 - - T
0,6
o
~
0,5
~
0,3
0,2
0,1
'"
x"
"" T = 67"C
0,4
:
EB-CO2 P=9MPa _
Re = 20 - 25
T
T"-"
f
0
0
(e)
ffi5 1 ° C /
300 600 900 1200 1500 1800 2100
t(s)
Fig. 8. Experimental (points) and predicted (lines)extraction
profiles at P = 9 MPa and Re = 20-25, for the systems: (a)
Bz/CO2/small pellets, (b) Tol/CO2/small pellets, (c)
EB/CO2/small pellets.
3538
F. Stfiber et al.
E
0,9 ; - iP=O'Olm/)
0,8
0,7
/
0,6
4
A
b*
dp = 0,02 m
g
/
A
A
A
F
0,5
0,4
/
D C B - CO2 -P = 20 MPa
0,2
t
2
T = 41°C
Re=37-41
0,1
t
0
t (s)
P=20
o%1
T
1000 2000 3000 4000 5000 6000 7000
(al
0,9
Benzene
QToluene
,LEthylbenzene
o Dichlorobenzene
x Lai and Tan, 1994
" ~ KnaffandSchlthnder,1987
ErkeyandAkgerman,1991/
. . . .
0,3
0
I
A
x
20
30
x
40
1
°6°~ t~ AIoo mA
50
T (°C)
60
70
80
Fig. 10. Variation of tortuosity factor of macroporous solid
with temperature (molecular diffusivities estimated with the
Catchpole-King (1994) correlation).
.~ ""
0,8
'~
0,7
/
'
rP=I2MP'
I
0,6
~
0,5
"~ 0,4
.
0,3
//
,
.
T = 40°C
R e - 14- 22
0
[
0
.
c.-co
0,2 1 /
0,1 ~ / ~ - - - -
Ib)
.
[
500 1000 1500 2000 2500 3000 3500
t (s)
Fig. 9. Experimental (points) and predicted (lines)extraction
profiles for the system DCB/CO2: (a) for two different sizes of
particle (1 and 2 cm equivalent diameter) P = 20 MPa and
Re = 37-41, T = 41~,2°C, (b) small pellets and different
pressures, T = 50°C, Re = 14-22.
were first reported by Knaff and Schlfinder (1987),
Recasens et al. (1989), Erkey and Akgerman (1990),
Madras et al. (1994), and Lai and Tan (1993). Table
7 summarizes specific data of their papers. As pointed
out earlier, the De obtained here are relatively higher
resulting in tortuosity factors from 0.2 to 0.5, although
D/De is still larger than unity. These tortuosity factors
are difficult to interpret. The results of Liang and Tan
with activated carbon where micropores dominate,
showing an extreme situation (D/De < 1), would
confirm the trend of our findings. Also, Knaff and
Schliinder (1987), working with a sintered bronze,
macroporous solid, similar to ours, found comparable
D/De in the range of 1-2 (with z between 0.4 and 0.5).
On the other hand, the data of Lee and Holder (1995)
and that of Erkey and Akgerman (1990) fall far above
our range, reporting both a value of r = 3.49, close to
the value one would expect from low-pressure, as well
as high-pressure heterogeneous catalysis (Zhou et al.,
1995). These large deviations between studies are not
easy to explain. These authors used supercritical adsorption chromatography, hence with some of the
common shortcomings of the moment method applied to dynamic data. In their chromatographic
method, five parameters (KA, ko, Dax, De, and kI ) are
fitted with two constraints arising from two moments.
Despite the fact that the precision of some of the
parameters, like D,x and De or kf , is questionable, the
problems associated with the moment method cannot
explain differences of De as high as 10 times. As an
example of the shortcomings associated with the
method we note the following. Lee-Holder and Erkey-Akgerman studied different porous solid matrices, (silica gel and alumina, respectively) but similar
operating conditions (bed length, bed diameter,
Reynolds number). The second authors postulated
that based on the calculations of kI (with the Lim
et a/.,1990 correlation), the external resistance is small
compared with intraparticle resistance. By contrast,
Lee-Holder obtained kf as well as De from their
measurements and developed a general equation in
terms of the Sherwood number. Despite their opposite
assumptions the two groups of authors report the
same value, r = 3.49.
With regards to the variation of z with temperature
and pressure, our results are somewhat scattered (see
Fig. 10 and Table 6). Table 6 shows that for the
Bz/CO2 system at constant pressure (9 MPa), the tortuosity factor increases by 50% when the temperature
is decreased by 15°C. The systems Tol/CO2 and
EB/CO2 show about the same trend at 9 MP pressure. For the three systems, Table 6 shows that the
Biot number is about the same (Bi = 13-15), well
above unity, indicating that intraparticle transport is
governing, hence the estimated diffusivity would be
reliable.
Supercritical fluid extraction
3539
Table 7. Survey of tortuosity factors for desorption or extraction of porous solids with super-critical carbon dioxide, and
experimental features
Authors
Conditions
Porous matrix
Tortuosity factor
Knaff and
Schlfinder
(1987)
Dissolution of naphthalene
from porous metal rod
T = 35-55°C
P = 12-22.6 MPa
Bronze, macropores
ep = 0.3
Pore size 8-20 pm
r independent of dp
and P
r = 0.40-0.54
Recasens et al.
(1989)
Modeling the
SCFE of ethyl acetate
(Tan and Liou, 1988 data)
T = 300-338 K, P < 13 MPa
Activated carbon
Internal and external
diffusion
LDV model
z~ 4
Erkey and Akgerman
(1990)
Naphthalene adsorption
P = %8-30.8 MPa
T = 298-318 K
Micro- and macroporous alumina
z const w/T and P
(z)av = 3.49
Lai and Tan (1993)
Adsoption of toluene
in a stirred basket-type
reactor
P = 7.7-16.3 MPa
T = 308-328 K
Micro- and macroporous activated
carbon
ep = 0.45
z-depends on pressure,
concentration
Desoption of heavy
organics
P = 10.0-11.3 MPa
T = 298-328 K
Soil, ep = 0.12
and activated
carbon, ep = 0.39
Micro- and macroporous solid
A mean value
(z)av = 3.0
Adsorption naphthalene
and toluene
P = 5.5-13 MPa
T = 301-318 K
Silica gel
(micro- and macropores)
A mean value
(z)av = 3.49
Extraction of 4 liquid
organics from porous
metal
P = 8-20 MPa
T = 305-330 K
Bronze, macropores
ep = 0.20-0.25
pore size 15/~m
z independent of P,
varies with T
z = 0.22q3.62
Madras et al., (1994)
Lee and Holder, (1993)
Present work
z = 0.18~0.6
LDV = Linear driving force model.
Erkey and Akgerman and Lee and Holder observed
no influence of pressure and temperature on z, and
gave an average value for the tortuosity factor
(Table 7) between 3 and 4. By contrast, the results of
Lai and Tan (1993) indicate a surprisingly strong
influence of pressure, and a modest effect of temperature (see Fig. 10). Our measurements for the case of
DCB are not conclusive, but tend to show the reverse,
i.e., an occasional effect of pressure and a larger effect
of temperature, in agreement with the findings
of Knaff-Schliinder (1987): refer to Table 6. F o r
example, the comparison of tortuosities at P = 12
M P a (and 42°C) with that at 20 M P a (and about the
same temperature, 41°C), would suggest a significant
increase of ~ with pressure (T from 0.48 to 0.62). But
comparison of the tortuosity at 12 M P a and T =
42°C, with that at P = 20 M P a and T = 39°C, does
not reveal an increase as before. Fig. 10 suggests,
however, that all our estimates are bounded by a tortuosity factor of about 0.62, with a minimum value of
0.21. Data by Knaff-Schli.inder (1987) and Lai and
Tan (1994) also fall in the same range. It is interesting
to note that Knaff-Schliinder and our values were
both for macroporous sintered bronzes (thus lending
confidence to each other's results), while Lai and Tan
used an activated carbon which is essentially microporous in nature.
The high values of De and the decrease of tortuosity
factor with increasing temperature, may give rise to
the assumption that surface diffusion contributes significantly to the general mechanism of diffusion in the
pores filled with dense gas (Knaff and Schliinder,
1987; Lai and Tan, 1993). In low-pressure gaseous
reactions on a porous catalyst, a typical sign of surface
diffusion is that the effective diffusivity exhibits
a negative temperature coefficient. That is, it is seen to
decrease with temperature. Smith (1981) has given
a reason for that. Assuming surface migration rate to
be at equilibrium with the pore fluid with respect to
the diffusing solute, the expression for the observed
effective diffusion coefficient is
De = (Dep + KaD~)/z
(31)
where it is assumed that ~ is the same in the pore
volume and on the surface migration path. Smith
3540
F. Stiiber et al.
(1981) notes that since adsorption is usually exothermic, K.4 will decrease with temperature, therefore De
will decrease with temperature too because the activation energy for Ds is small, while the enthalpy change
for adsorption is much larger and negative. However,
for the case of adsorption from supercritical carbon
dioxide, data show that for certain range of pressures,
adsorption isotherms are reversed with respect to
temperature. Therefore, the adsorption process is
endothermic and hence retrograde. The pressure
range where this occurs is for pressures below the
crossover pressure of the system (Recasens et al., 1989,
1993). If this is the case, when pores are filled with
SCF, the second contribution of the rhs of eq. (31) will
tend to increase at higher temperatures. Then the total
diffusivity will increase with temperature faster than
molecular diffusivity, giving rise to a tortuosity factor
that would appear smaller at larger temperatures.
This effect is that observed in Fig. 10 (see bottom data
points) where tortuosity factors decrease with increasing temperatures for a series of runs.
In addition to proving or disapproving surface diffusion, there is still the fact that tortuosity factors are
less than unity. For catalytic gaseous reactions at low
pressure, the authors agree that tortuosity factors are
in the range 1-10, in which ~ = 3-4 would be the most
typical. Smith (1981) indicates that the occurrence of
tortuosity factors smaller than unity is when surface
diffusion is significant. In fact, many low-pressure
gas-liquid reactions on a catalyst whose pores are
filled with a dense fluid (not necessarily supercritical,
but just a liquid) exhibit a tortuosity factor less than
unity. For example, the hydrogenation of :t-methylstyrene on Pd/Alumina in a trickle-bed reactor
(Herkowitz et al., 1979) or the adsorption of nitric
oxide or hydrogen sulphide in aqueous slurries of
activated carbon (Niiyama and Smith, 1976), etc.
Ramachandran and Chaudhari (1983) give a short
survey on tortuosity factors found in multiphase
reactor operation. Only in low-pressure reaction or
adsorption with gas-limited, liquid-filled pores tortuosity factors smaller than unity are found.
The explanations above are speculative. Probably,
in the light of existing data, more specific investigation, particularly a check on the presence of pore
adsorption itself and modelling surface diffusion, is
a pre-requisite to explain and understand the physical
mechanisms involved.
CONCLUSIONS
The supercritical-fluid extraction of organics (benzene, ethylbenzene, toluene, dichlorobenzene) from macroporous sintered-metal cylinders, either with sealed
ends or permeable ends, has been investigated both
experimentally and theoretically, using the shrinkingcore model and the differential-reactor approximation,
Extraction runs were of two types. (1) Those performed at quasi-equilibrium in which the fluid became
saturated with the organics. These experiments allowed to measure solubilities of the organics in the
supercritical fluid. Solubilities were modelled with the
Peng-Robinson equation of state, in terms of adjustable binary interaction parameters (klz). (2) Further
extraction runs carried out at a higher flowrate, in the
range of Re ~20-50, hence under conditions influenced by mass-transfer. Fitting the latter runs with
the shrinking core model (with the external mass
transfer coefficients available from a previously derived correlation) provided estimates of effective diffusivities, which could be compared with molecular
diffusivities. The resulting tortuosity factors were
found to be less than unity, and to decrease with
temperature, a feature exhibited as well by gas-limited
catalytic reactions in liquid-filled pores in slurry- and
trickle-bed type of multiphase reactors.
Acknowledgments
The Ecole des Mines d'Albi (France) and CIRIT (PIEC
programme, Generalitat de Catalunya, Barcelona), respectively, kindly provided fellowships to Mr Strphan Julien and
to Dr Frank Stiiber, for work in this laboratory. The help of
AMES (Sant Vicenq dels Horts, Barcelona) in supplying
samples of sintered metallic parts and experimental methodologies, is appreciated. The project has been supported by
the Spanish Ministry of Education and Culture (Madrid,
CICYT Grant Nr. AMB95-0042-C02-02).
aeff
B
b
Bi
C
m
Cg
c.*
CL
D
De
Ds
deq
Gr
K
Ko
k12
L
l
lc
N
P
qo
NOTATION
effective specific mass transfer area,
m2/m 3
bed length, m
factor in eq. (24)
Biot number, ( = KoR/De )
concentration, C9 = fluid phase, Ci =
within the fluid-filled pores, mol/m 3
mean fluid-phase concentration over a
time interval, mol/m 3
concentration at saturation in fluid
phase, mol/m 3
concentration of solute in the liquid,
mol/m 3
molecular binary diffusivity at infinite dilution, m2/s
effective diffusivity, m2/s
surface diffusivity,
Particle diameter, m
diameter of equivalent sphere, m
Grashof
number
for
mass,
[ = (dapg/v)(Ap/p)]
partition coefficient, ( = Cg*/CL)
fluid overall mass transfer coefficient, m/s
binary interaction parameter of PengRobinson equation of state
length of cylinder, m
coordinate of flat end relative to cylinder
centre, m
length of cylindrical core, m
amount of substance, mol
pressure, Pa
initial solute concentration based on
solid volume, mol/m 3
Supercritical fluid extraction
R
r
rc
Re
Sc
Sh
t
T
u
v
X
y
pellet radius, m
radial coordinate within body, m
core radius in the shrinking-core model, m
Reynolds number, (= pgvdeq/p)
Schmidt number (= v/D)
Sherwood number, (=Kgdeq/D, ShN,
Shr, Shr, Sho) stand for natural convection, total, forced and for molecular
transport Sherwood number
time, s
temperature, K
insterstitial velocity, m/s
superficial velocity, m/s
fraction extracted in a differential bed
mole fraction of organic in vapour phase
Greek letters
eB
bed void fraction
ep
particle porosity
p
fluid density, kg/m 3
Ap
absolute difference (= IP* - pD, kg/m3
p*
fluid density when it is saturated, kg/m 3
v
dynamic viscosity, m2/s
z
Tortuosity factor, ( = e,pD/De)
Subscripts and superscripts
A
adsorption
a
adsorption
ax
axial
f
fluid-particle
C
core
e
g
L
p
0
*
1
2
effective
fluid
liquid
particle
initial
saturated solution
carbon dioxide
solute
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